MAM Seminars Autumn term 2015

Higher seminars in the subject Mathematics/Applied Mathematics, Autumn term 2015. School of Education, Culture and Communication (UKK), Mälardalen University.

Program for Mathematics and Applied Mathematics seminar.

Autumn term 2015

Wednesdays afternoon is the normal time for MAM seminars with deviations when necessary. The program is always provisional. The information about each specific talk at MAM seminar becomes final the day before. Suggestions for talks at MAM seminar are very welcome to Prof. Sergei Silvestrov sergei.silvestrov@mdh.se.

Many special functions find wide application in Number Theory and Physics: through a no-conventional approach (using Fractional Calculus, Wavelet Analysis and Fractal Geometry), we can describe a family of them. It is well know that almost all special functions are functions of complex variable: Ortigueira, defines a generalized Caputo derivative for complex functions with respect to a given direction of the complex plane: so the fractional derivative of a Dirichlet series, Hurwitz zeta function and Riemann zeta function can be easily computed as a complex series. Some interesting properties of the fractional derivative of the Riemann zeta function are also investigated to show that there is a chaotic decay to zero.

A general theory is developed for the eigenvalue effective size (NeE) of

structured populations in which a gene with two alleles segregates in discrete time.

Generalizing results of Ewens (Theor Popul Biol 21:373–378, 1982), we characterize NeE in terms of the largest non-unit eigenvalue of the transition matrix of a Markov chain of allele frequencies.We use Perron–Frobenius Theorem to prove that the same eigenvalue appears in a linear recursion of predicted gene diversities between all pairs of subpopulations. Coalescence theory is employed in order to characterize this recursion, so that explicit novel expressions for NeE can be derived. We then study NeE asymptotically, when either the inverse size and/or the overall migration rate between subpopulations tend to zero. It is demonstrated that several previously known results can be deduced as special cases. In particular when the coalescence effective size NeC exists, it is an asymptotic version of NeE in the limit of large populations.

Universal enveloping algebras of Lie algebras is one of the most classical families of noncommutative associative algebras. One of their notable properties is that they can be regarded as algebras of ordinary (commutative) polynomials (i.e., they are as vector spaces isomorphic to algebras of commutative polynomials) endowed with a different noncommutative multiplication operation, something which is called the Poincaré–Birkhoff–Witt (PBW) property. In this talk, I will show that a real-valued function on a basis of a Lie algebra can be extended to a valuation on the universal enveloping algebra of that Lie algebra, provided it satisfies a natural and quite generous compatibility condition. The proof relies on a result that the universal enveloping algebra of a Lie algebra can be embedded into a power series algebra which (i) satisfies a PBW-type property and (ii) is a skew field.

on the same interval. Clearly the solution f of (2) is nonnegative. We studied the problems of optimizing the area under the function f on the interval [0,T], when r is allowed to vary over (i) all its equimeasurable rearrangements (ii) the convex hull of all its equimeasurable rearrangements.

We give a new and very short proof of the infimum problems. We also set conditions characterizing the solvability of the supremum problems.

October 7, 2015, Wednesday, 15.30-16.30

Location: U3-083 (Hilbert room), Västerås, UKK, Mälardalen University

Speaker: John Bondestam Malmberg, Department of Mathematical Sciences, Chalmers University of Technology and University of Gothenburg

Title:

A Two-stage Numerical Procedure for an Inverse Scattering Problem for Maxwell's Equations

Abstract:

We study a numerical procedure for the solution of the inverse problem of reconstructing location, shape and material properties (in particular refractive indices) of scatterers located in a known background medium. The data consist of time-resolved backscattered radar signals from a single source position. This relatively small amount of data and the ill-posed nature of the inversion are the main challenges of the problem. Mathematically, the problem is formulated as a coefficient inverse problem for a system of partial differential equations derived from Maxwell's equations.

The numerical procedure is divided into two stages. In the first stage, a good initial approximation for the unknown coefficient is computed by an approximately globally convergent algorithm. This initial approximation is refined in the second stage, where an adaptive finite element method is employed to minimize a Tikhonov functional. An important tool for the second stage is a posteriori error estimates -- estimates in terms of known (computed) quantities -- for the difference between the computed coefficient and the true minimizing coefficient.

Some Applied Aspects of fixed point results in Generalized Metric Spaces

Abstract:

Inspired from the impact and utility of metric space, several generalizations of this notion have been introduced in the literature such as semi-metric space, fuzzy metric space, dislocated metric space, probabilistic metric (Menger) space, G-metric space, cone metric space. The fixed point theory as a part of non-linear analysis is a study of function equation in metric or non-metric setting. It provides necessary tools for the existence of theorems in non-linear problems. The classical Banach contraction principle is one of the fundamental results in metric space with wide applications. This principle has also a vital role on establishing fixed point results for non-expansive mappings in Banach and Hilbert spaces. The main purpose of this presentation is to discuss some developments of classical metric subspaces and contractive definitions with applications to other disciplines.

We study a problem of pricing a contract consisting of two perpetual American options between two defaultable counterparties. We show that the problem reduces to an optimal stopping game. Moreover, we show that a Nash equilibrium as well as the value of the game exist and we state and prove a verification theorem. Then, we calculate Credit Valuation Adjustment (CVA) of this contract. This paper can be considered as the first step in pricing larger volumes of contracts between two defaultable counterparties.

I will describe a generalization of certain results by Hellström and Silvestrov on centralizers in graded algebras. More precisely I have studied centralizers in certain algebras with valuations. I have proven that the centralizer of an element in theses rings is a free module over a certain ring. Under further assumptions we obtain that the centralizer is also commutative.

Reducibility of the wavelet representations associated to the Cantor set

Abstract:

This work is devoted to interplay between wavelets, fractals, operator theory and representations. An introductory review of the article on the subject by Sergei Silvestrov and Dorin Dutkay and some related works will be given.

Fractal antennas represent a surprising application of the theory developed by Mandelbrot to antenna theory. The two fundamental properties of a fractal (i.e. self-similarity and space-filling) allow these antennas to have an efficient miniaturization and an excellent multiband behaviour. Fractal antennas have different fields of application, like defence, cultural heritage conservation and spatial communications: in addition, they may be used to obtain the so-called broadband invisibility cloak.

Enhancement of Impedance Bandwidth of Equilateral Triangular Microstrip Antenna Using Different Feeding Techniques

Abstract:

Novel design of equilateral triangular microstrip antenna is proposed at X-band frequency. The antenna is designed, fabricated and tested for single and multiband operation. The experimental impedance bandwidth of conventional equilateral triangular microstrip antenna (CEMA) is found to be 5.02%. The CEMA is simulated using HFSS software.The study is continuedwith array technique and the radiating patches are fed by corporate feeding technique. An optimized wide slot is inserted at the centre of radiating patches of array antennas to enhance the impedance bandwidth and other antenna parameters. It is observed from the results that the impedance bandwidth of proposed antennas is enhanced up to 9.11 % which is 80.08% more when compared to the impedance bandwidth of CEMA. Theoretical results of the proposed antennas are well agreed with the experimental results. Input impedance, gain, azimuthal radiation patterns and VSWR of proposed antennas have been studied and reported. The obtained experimental results and theoretical study of the proposed antennas are given and discussed in detail.

Mapping class groups of surfaces constitute a class of infinite discrete groups that is on the one hand interesting for a number of applications and on the other hand notoriously difficult to get a good handle on. Some theories from Physics describing low-dimensional systems produce finite dimensional representations of mapping class groups which have proved useful to investigate properties of mapping class groups themselves. I will discuss recent results (obtained in collaboration with Jürgen Fuchs) partly characterising a certain class of such representations constructed from any finite group G. The class of representations that we consider is a generalisation of certain representations of the modular group appearing in finite group theory (in work by Drinfeld) as well as in the so-called Moonshine programme.

December 16, 2015, Wednesday, 15.30-16.30

Location: U3-083 (Hilbert room), Västerås, UKK, Mälardalen University

Speaker: Urban Larsson, Dalhousie University, Halifax, Canada

Title:

Self-organization in combinatorial games

Abstract:

By generalizing the classical heap game of Wythoff Nim (a.k.a Corner-the-Lady), we have various empirical evidence of self-organization of the generalized outcomes. Moreover, a new diamond shaped Cellular Automaton emulates these patterns. This is joint work with Matthew Cook, Turlough Neary, Adam Landsberg, Scott Garrabrant, Eric Friedman and Ilona Phipps-Morgan.